Anyway, the equation
oil extraction: (capital) + (skilled labour) + (machinery) + (oil fields) -> (crude oil)
(oxygen) + (hydrogen) -> (water)
O<sub>2</sub> + 2 H<sub>2</sub> -> 2 H<sub>2</sub>O
One of the interesting insights that allowed Leontief to make a useful model is that he reversed the flow of the "production" equations/processes. Instead of asking "how much of the output can be produced given the inputs?" which, if the quantities are not right will result in some of the inputs being left over, he asked "how much of (input) is needed to produce one unit of (output)?". This is the inverse of the productivity of the input, by the way. The cool thing is that, given the inputs, how much and which outputs arre produced is an indeterminate problem, and not all the inputs may end up being used; but, given the outputs, you know exactly how much of the inputs has been used.
This is why actually none of the equations you has written really is a correct interpretation of how to turn the "process equations" into linear equations. But, then again, all this is true of chemical reaction equations. The symbols '+', '->' and the coefficients have the same meaning. The 1/2 in
(1/2) O<sub>2</sub> + H<sub>2</sub> -> H<sub>2</sub>O
So, there is a linear model at the end, but it is not obvious from the presentation of the "process" equations, not even if the "productivity" (stoichiometric) coefficients are given.
As an aside, I think this is clear why this was an insane choice of ancillary topic in the course I was assisting. Maybe there is a diary in there where I pick apart college level mathematics education in the US.
Anyway, the assumption of linearity is implicit in the question "how much of (input) is needed to produce one unit of (output)?", but you can of course replace that with a nonlinear function. I am more interested, however, in the "process equations", which might be represented as a directed graph (with nodes representing processes and arrows representing inputs and outputs). Those whom the Gods wish to destroy They first make mad. -- Euripides
I am more interested, however, in the "process equations", which might be represented as a directed graph (with nodes representing processes and arrows representing inputs and outputs).
H'mmmm.
What you need is a Neural Net.
If the model isn't linear then presumably it's pretty much unstable, almost by definition. I suppose a subset of small problems might convege towards something resembling equilibrium. But won't a reality-sized model that assumes time-based relationships between inputs and outputs wobble chaotically all over the place?
I agree, though, that as soon as you introduce time and, more importantly, delays, you get an entirely different order of complexity. Delayed feedback is really, really nasty in that respect. Those whom the Gods wish to destroy They first make mad. -- Euripides
Perhaps I should actually break down and look at Leontief's methods. But since I heard that he treated "technology" as a scalar input, I've had a disgust toward the topic.
...Hmmm... Google doesn't think that "process equations" have much to do with "Leontief" (10 hits)....Further digging suggests that Leontief was indeed saying what you and I have trouble believing he'd say:
The neoclassical production function is an explicit function Q = f(K,L), where Q = Quantity, K = Capital, L = Labor... Leontief, the innovator of input-output analysis, uses a special production function which depends linearly on the total output variables.... Widipedia: Input-output model
Q = f(K,L),
where Q = Quantity, K = Capital, L = Labor... Leontief, the innovator of input-output analysis, uses a special production function which depends linearly on the total output variables....
Widipedia: Input-output model
I'm puzzled. Words and ideas I offer here may be used freely and without attribution.
The inimitable book by Leontief himself remains the best exposition of input-output analysis. See bibliography.
The wikipedia article also says
Despite the clear ability of the input-output model to depict and analyze the dependence of one industry or sector on another, Leontief and others never managed to introduce the full spectrum of dependency relations in a market economy.
Irving Hock at the Chicago Area Transportation Study did detailed forecasting by industry sectors using input-output techniques. At the time, Hock's work was quite an undertaking, the only other work that has been done at the urban level was for Stockholm and it was not widely known. Input-output was one of the few techniques developed at the CATS not adopted in later studies. Later studies used economic base analysis techniques.
In 2003, Mohammad Gani, a pupil of Leontief, introduced Consistency Analysis in his book 'Foundations of Economic Science', which formally looks exactly like the input-output table, but explores the dependency relations in terms of payments and intermediation relations. ... In a technical sense, input-output analysis can be seen as a special case of consistency analysis without money and without entrepreneurship and transaction cost.
...
In a technical sense, input-output analysis can be seen as a special case of consistency analysis without money and without entrepreneurship and transaction cost.
